Mini Lecture.1 Introduction to Functions 1. Find the domain and range of a relation.. Determine whether a relation is a function. 3. Evaluate a function. 1. Find the domain and range of the relation. a. {(1, 5), (, 10), (3, 15), (, 0), (5, 5)} b. {(1, -1), (0, 0), (-5, 5)}. Determine whether each relation is a function. ( 5, 6), (5, 7), (5, 8), (5, 9), (5,10) b. {(5, 6), (6, 7), (7, 8), (8, 9), (9, 10)} a. { } 3. Find the indicated function value. a. f ( 3) for f ( x) = 3x b. g ( ) for g( x) = x x + c. h ( 1) for h( t) = t 3t + d. f ( a + h) for f ( x) = x + 3. Function g is defined by the table x g (x) 0 1 6 3 8 10 Find the indicated function value. a. g() b. g () Teaching Notes: A relation is any set of ordered pairs. The set of all first terms x-values of the ordered pairs is called the domain. The set of all second terms y-values of the ordered pairs is called the range. A function is a relation in which each member of the domain corresponds to exactly one member of the range. A function is a relation in which no two ordered pairs have the same first component and different second components. The variable x is called the independent variable because it can be assigned any value from the domain. The variable y is called the dependent variable because its value depends on x. The notation f(x), read f of x represents the value of the function at the number x. Answers: 1. domain {1, 0, -5} range {-1, 0, 5}. a. not a function b. function 3. a. 7 b. 1 c. 6 d. a + h + 3 n. a. 6 b. 10 ML-10 Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall
Mini Lecture. Graphs of Functions 1. Graph functions by plotting points.. Use the vertical line test to identify functions. 3. Obtain information about a function from its graph.. Identify the domain and range of a function from its graph. State the domain of each function. 1. Graph the function f ( x) = 3x and g( x) = 3x + 1 in the same rectangular coordinate system. Graph integers for x starting with and ending with. How is the graph of g related to the graph of f?. Use the vertical line test to identify graphs in which y is a function of x. a. b. c. 3. Use the graph of f to find the indicated function value. a. f() b. f(0) c. f(1) 6 Y X -6 - - 0-6 - -6. Use the graph each function to identify its domain and range. a. b. Y 6 6 (-5,1) (1, 1) (-, 1) (, 1) X X -6 - - 0-6 -6 - - 0-6 - - -6-6 Y ML-11 Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall
Teaching notes: The graph of a function is the graph of the ordered pairs. If a vertical line intersects a graph in more than one point, the graph does not define y as a function of x. Answers: 1. The graph of g is the graph of f shifted up 1 unit.. a. yes b. no c. yes 3. a. 0 b. c. 1. a. Domain: {x x = -5, -, 1, } Range: {y y = 1} b. Domain: {x x } Range: {y y is a real number} ML-1 Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall
Mini Lecture.3 The Algebra of Functions 1. Find the domain of a function.. Use the algebra of functions to combine functions and determine domains. State the domain of each function. = x 5 x + = x + 6 1. a. f ( x) b. g ( x) c. h ( x) = x + d. p( x) = x x. a. g ( x) = x b. h ( x) = x 7 c. b ( x) = x + 3 d. m ( x) = x 8 3. Let f ( x) = x x and g ( x) = x + 3 a. ( g)( x). Find the following; f + b. the domain of g 5 6 =. Find the following; x f + b. The domain of f + g. Let f ( x) = and g( x) x + a. ( g)( x) f + c. ( f + g)( ) 5. Let f ( x) = x + 1 and g ( x) + x = 3. Find the following; a. ( f + g)( x) b. ( f + g)( ) c. ( f g)( x) f g d. ( f g)( 0) e. ( ) Teaching Notes: Students need to be reminded that division by zero is undefined. The value of x cannot be anything that would make the denominator of a fraction zero. Students often exclude values from the domain that would make the numerator zero, warn against this. Show students why the radicand of a square root function must be greater than or equal to zero. This is a good place to use the graphing calculator so students can see what happens. Answers: 1. a. {x x is a real number and x 5 } b. {x x is a real number and x 6 } c. {x x is a real number} d. {x x is a real number and x 0 }. a. { x x 0} b. { x x 7} c. { x x -3} d. { x x is a real number} 5 6 3. a. x w x + 3 b. { x x is a real number} c. 9. a. + x + x b. {x x is a real number and x } and x 0 } 5. a. x + x b. 0 c. x x + d. e. -1 ML-13 Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall
Mini Lecture. Linear Functions and Slope 1. Use intercepts to graph a linear function in standard form.. Use the algebra of functions to combine functions and determine domains. 3. Graph linear functions in slope-intercept form.. Graph horizontal and vertical lines. 1. Use intercepts and a checkpoint to graph each linear function. Name the x-intercept and the y-intercept. a. x + 5y = 10 b. x y =. Find the slope of the line passing through each pair of points. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. a. (, 5) and ( 6, 3) b. ( 5, 0) and (1, 3) 3. a. Find the slope and y-intercept for the line whose equation is 3 x + y = 1 and then graph the equation. 1 b. Find the slope and y-intercept for the linear function f ( x) = x + 3 and then graph the function.. Graph the linear equations. a. x = b. 3 y = 1 5. c. ( f g)( x) d. ( f g)( 0) e. ( ) 6. c. ( f g)( x) d. ( f g)( ) ML-1 f g Teaching Notes: The standard form of the equation of a line is Ax + By = C, as long as A and B are not both zero. A x-intercept will have a corresponding y coordinate of 0. A y-intercept will have a corresponding x coordinate of 0. rise The slope of a line compares the vertical change to the horizontal change ( ) run Slope formula is: y y 1 m =. x x1 A line that rises from left to right has a positive slope. A line that falls from left to right has a negative slope. A line that is horizontal has zero slope. A line that is vertical has an undefined slope. Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall.
The slope-intercept form of the equation of a line is y = mx + b where m is the slope and b is the y-intercept. Answers: 1. a. b. 1 3 3. a. m =, rises b. m = or, falls 3 3 3. a. y = x + 3 m = y -intercept (0, 3) b. 1 m = y-intercept (0,3). a. b. 5. c. x x + d. e. -1 6. c. x 5x 3 d. 1 ML-15 Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall
Mini Lecture.5 The Point-Slope Form of the Equation of a Line 1. Use the point-slope form to write the equation of a line.. Find slopes and equations of parallel and perpendicular lines. 1. Write an equation in point-slope form for the line with the given information. a. slope 3 passing through (, 1) b. slope 3 passing through (6, 3) c. slope 0 passing through ( 3, ). Write an equation in point slope form for the line with the given information. Then write the equation in slope-intercept form. a. slope 1 passing through (, 3) b. passing through (1, 5) and (3, 5) c. passing through (, ) and (, 6) 3. Find the slope of a line parallel to each given line. 3 a. y = x + 7 b. x 3y = 6 c. x + y = 8. Find the slope of a line perpendicular to each given line. a. y = x + 1 b. 3 x + y = 1 c. x y = 10 Write an equation for each of the following in slope-intercept form. 5. A line passing through the origin (0, 0) and parallel to a line whose equation is y = x 5. 6. A line passing through (, ) and parallel to a line whose equation is x + y = 6. 7. A line passing through (6, 1) and perpendicular to a line whose equation is x 3y = 1. 8. A line passing through (, 5) and perpendicular to a line whose equation is y x + 7 = 0. Teaching Notes: The point-slope form of the equation of a non-vertical line with slope m that passes through the point ( x, y) is y y1 = m( x x 1 ). Make sure students memorize the point-slope form of a linear equation and know what each letter represents. Students will need lots of practice on this. ML-16 Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall
This is a good time for students to be able to visualize parallel line lines and see that they have the same slopes, but different y-intercepts. Answers: 1 1. a. y 1 = 3( x + ) b. y 3 = ( x 6 ) c. y = 0( x + 3). a. y 3 = ( x ) ; 3 1 1 y = x + b. y 5 = 5( x 1) or y + 5 = 5( x 3) ; y = 5x + 10 c. y 6 = ( x ) or 1 1 3 1 1 1 y = ( x + ) ; y = x + 5 3. a. b. c.. a. b. c. 5. y = x 3 1 3 1 6. y = x 7. y = 3x + 17 8. y = x 1 ML-17 Copyright 01 Pearson Education, Inc. Publishing as Prentice Hall